## How I Got Here

I believe that space is built up out of tiny granules rather like the pixels on a TV screen. I care deeply about it. I also think that every software developer, computer scientist, and discrete mathematician in the world should hold the same opinion and be working night and day to find out how that kind of universe might work.

This is because I’m convinced that unless the idea of discrete space is investigated, humanity will never uncover the secrets of the universe. Progress at the frontiers of science will die. Our species will enter a duller, more closed-minded age, and the process of enlightenment that started with the Renaissance will have fizzled out without us finishing the job. I’m hoping to convince you to join me, and work with me on a crusade to revolutionize science and to keep it alive and healthy for the next five hundred years.

Before we begin, though, it’s important to point out that the vast majority of professional physicists reject discrete space models after dedicating years of study to the topic. Furthermore, the current experimental evidence supporting the hypothesis is thin and circumstantial. And all the best tools that physics has for explaining the universe are fundamentally smooth. They’re based on calculus–Newton’s masterpiece of mathematics–a system that has proven phenomenally, unbeatably successful over the last three hundred years.

How can I possibly hold such a position, then, let alone hope to convince you of it? The answer will probably require multiple posts, as well as clear outlines as to what I’ve already achieved. Nevertheless, we must start somewhere. So let’s start with a story.

I didn’t always believe that space was digital. I was convinced by the one man in the world who probably dislikes the idea of discrete space more than anyone else: Sir Roger Penrose. Roger Penrose is the genius who invented quasi-periodic tiling and wrote the bestseller pop-science book The Road to Reality. Long before he wrote that, though, he wrote The Emperor’s New Mind.

I encountered the work of Roger Penrose in my late twenties. I’d started my career as an Artificial Intelligence researcher then spent a few years dabbling with being a full-time writer of science fiction novels. During that time I happened to be lucky enough to attend Clarion West–probably the most intensive and prestigious science fiction writing program in the world. While I was there, I wrote a novel outline based on the idea of human minds being uploaded into computers, a very ‘du jour’ idea at the time, and one where my background could help the plot. The outline went down well with my critique group except for with one friend of mine.

“It’s a nice idea, Alex,” she said. “Shame it could never happen.” I confess I was somewhat taken aback by this response and asked her why. She told me that Roger Penrose had explained in Emperor’s New Mind that consciousness worked on principles that computers could never capture. When I asked her to explain further, she waved her hand and said that she didn’t really understand it fully. I’d have to read the book to find out why.

As soon as the workshop was over, I picked up a copy of Emperor’s New Mind and started dissecting it. Penrose’s argument, in essence, was that consciousness wasn’t like a computer program because it relied on phenomena in the universe that weren’t like computer programs either. He then went on to describe all those phenomena that he believed nailed the case that the universe could never be expressed in algorithmic form.

I found myself going over each example carefully and in every case came away convinced that Professor Penrose was *exactly wrong*. Not only was consciousness like a program, but *everything else* was too. This result astonished me. Prior to that moment, I hadn’t given the idea all that much thought. Now, though, I had a passion for the subject, and launched myself on a voyage of research and discovery. It would be another eleven years before I published my first scientific paper on the subject. I’m now working on my third and have spoken at two international conferences on the subject.

In this blog, I’m hoping to share some of the fruits of my work. Once I’ve covered some of the philosophical foundations, I’ll share with you some of the algorithms I’ve discovered, and also talk about the work of some of the brilliant people I’ve encountered along the way. I’m still convinced that Penrose was fabulously, extravagantly wrong, but now I can go further. I can state, with some confidence, I think, that believing in discrete space is the *same thing as believing in science*. To believe otherwise, despite appearances, constitutes a forgivable, if lamentable, form of mysticism. In the next post, I’ll explain why.

Wow, Alex. This sounds quite impressive. I am somewhat minded of Rudy Rucker’s recent work. Did you read “The Lifebox, the Seashell, and the Soul?”

I await your forthcoming posts eagerly.

I’m looking forward to how this pans out. I’ve always understood Penrose to be spectacularly wrong about many things, but consciousness was right at the top. Quantum microtubules, schmantum microtubules.

Does Wolfram’s book on discrete automata touch on this area? It seems very likely that the ‘smoothness’ that we see all through nature is easily manufacturable from simple discrete feedback systems. Looks at genes and organic life for even a microsecond. Can this stretch to physics as well?

But then again, I’m also aware that my experiences programming computers gives me an increasingly illusory understanding of the world. When we make simulations of what we see around us, we have to make everything discrete, because this is the only paradigm computers understand. We approximate the smoothness. And when we get up from doing that, all the smoothness we see around us makes us think of those discrete functions we’ve been tinkering with, and apply them backwards. When you’ve got a powerful hammer, everything looks like a nail.

Ever since I first heard you talking about this idea in Cambridge, I’ve been thinking about it too. But I could never get past my first thought, which concerned photons coming from the Sun. If they travel 93 million miles, and I can receive photons from it across every millimetre of my pupil, then the angle subtended at the sun, when those photons set out, would be so infinitesimally small to make any ‘quantum positioning’ argument moot. Wouldn’t it? I’m hoping I’ve misunderstood you…

But it’s all worth investigating. Where can I sign up? And can we read your papers please?

If you want to read my papers, drop me a line at alex dot lamb at gmail dot com I’ll forward you a couple of preprints. I got the galley proof from Complex Systems through for paper one today, which is I’m very excited about. With luck it’ll hit print soon.

As for the modeling the quantum behavior of spreading and interfering waves in a discrete system, watch this space. I have some very exciting results that haven’t made it into a paper yet. 🙂

Wolfram has a whole chapter on physics that I found somewhat compelling… He sort of stops halfway, and knowing SW my suspicion is he’s planning to copyright and patent the TOE. The TOE will be monetized! We’ll have to pay a license fee to apply the basic laws of physics. Hey, if they can patent genes…

As to your comment about the angle of a photon — well that’s a pretty loaded and tricky question, and gets to the heart of that whole wave/particle duality problem. I have some ideas (as does Alex) — but to begin with, I would point you to the Feynman path integral model as a source of good ideas about the matter. Turns out the likely path of a photon (or any particle) can be thought of as the sum of a potentially infinite series of possible paths. (however I think this sum is calculated from something called a probability amplitude, which is squared to get the actual probability — meaning there are something analogous to negative probabilities involved).

In any case, there are plenty of such thought experiments that seem to make the idea of a computable Universe difficult to formulate with any precision. I think most of this comes from the fact that we have historically used the calculus of continuous variables as the foundation for our theories. It is impossible in principle to digitally simulate a set of differential equations with perfect accuracy. You must choose a level of precision — a ‘gauge’ as it is sometimes called — at which you will perform the numerical recipes that approximate the actual equations to varying degrees.

But if you turn the situation upside down, and assume for a moment that the reason calculus serves us well is *precisely* *because* we do not know the actual, exact discrete theory — then it makes sense that we have used math up to now, because it gives us the most predictive utility with the least knowledge of the underlying process.

Our task, then, is qualitatively different than has been the case in the past. Rather than find better and better approximations to an unknowable secret, we wish to reveal the secret itself. It’s more like searching for a cryptographic key to break an encryption scheme than it is like trying to find a function that fits a set of data.

I hope this makes some sense. If not, we will keep trying to make sense until we all feel sensible about this.

-dan miller

Hi Dan,

Intriguing stuff there. From what I remember of Feynman’s book on QED, he said something along the lines of “It’s AS IF the particles travel in every possible route to get to the point you’re interested in, and if you accept that they do this, and do the calculations, then it all turns out perfectly, and the math works. It may or may not be happening that way – who knows – but as long as it works, that’s how we do those calculations”.

He was a bit of an instrumentalist (being at heart, I think, an engineer) but I think he realised that a number of interpretations fit the QED model well, without any of them necessarily being any closer to the others to ‘The Truth’.

There are lots of engineering problems whose solutions lie in the solutions to horribly complex integral and differential equations, but advances in mathematics have rendered those problems soluble by non-iterative means (i.e. algebraically). When worked out on a piece of paper, they yield a simple formula which can give any precision you like, to as many decimal places as you like, which far outstrips the necessarily iterative ability of computers.

I’m not sure if I’ve understood you right about ‘revealing the secret’. I’m fairly sure I don’t understand the assumption in your ante-penultimate paragraph! I can’t see, just because we can’t get perfect digital representation of infinite series, that it means a digital foundation is any more likely.

My brain is slowly coming round to various ideas in this realm, though – been screening through Wikipedia’s relevant topics.

I’ll send you an email, Alex.

Cheers,

-C

Christopher —

> [re Feynman] He was a bit of an instrumentalist (being at heart, I think, an engineer) but I think he realised that a number of interpretations fit the QED model well, without any of them necessarily being any closer to the others to ‘The Truth’.

I suspect that describes the majority of physicists who made significant, concrete contributions to the field (as opposed to those whose work tends more to the ‘angels at the end of a string’ variety). The archetypal attitude here is “shut up and calculate”.

> There are lots of engineering problems whose solutions lie in the solutions to horribly complex integral and differential equations, but advances in mathematics have rendered those problems soluble by non-iterative means

That’s certainly only true in specific cases, not in the general case. Something as simple as 3 bodies with mutual attraction is still not solvable by non-iterative means, right? So a world composed of some 10^80 distinct entities, all with the potential to attract, repulse, or otherwise interact with each other, is not likely to be solvable in any sense other than through numerical simulation, at some level of precision.

> I can’t see, just because we can’t get perfect digital representation of infinite series, that it means a digital foundation is any more likely.

Alex may be better suited to outline the metaphysical arguments regarding this proposition. The simplified version goes something like this: since a perfect answer to the result of an infinite series requires (in the most general case) infinite calculation; whereas a reasonably good discrete approximation using finite methods would be indistinguishable experimentally; by Occam’s razor, we should prefer the hypothesis that the Universe is discrete, as it is in some sense the simpler solution.

I’ve not done the argument justice unfortunately. In any case, from my perspective, there are other justifications for pursuing the discrete theory. At the smallest scale, we see a discretization of energy transfer in the quantum. At the other end, we see C as a limit on information transfer. To my mind, these are strange artifacts for a universe based on the platonic continuum. Why not just make it Newtonian? Why limit the amount of information involved in energy exchange, or the speed at which that information can disperse? If you postulate infinite computational ability in the physical substrate, those limitations seem like glaring inconsistencies. To me at least.

-dan

Regarding Feynman, I found these two provocative quotes from the (recently revamped) Wikipedia article on digital physics:

[quoting from http://en.wikipedia.org/wiki/Digital_physics as of 2/27/10:]

As Richard Feynman put it:

“It always bothers me that, according to the laws as we understand them today, it takes a computing machine an infinite number of logical operations to figure out what goes on in no matter how tiny a region of space, and no matter how tiny a region of time. How can all that be going on in that tiny space? Why should it take an infinite amount of logic to figure out what one tiny piece of space/time is going to do?”[26]

He then answered his own question as follows:

“So I have often made the hypothesis that ultimately physics will not require a mathematical statement, that in the end the machinery will be revealed, and the laws will turn out to be simple, like the checker board with all its apparent complexities. But this speculation is of the same nature as those other people make—’I like it,’ ‘I don’t like it’—and it is not good to be prejudiced about these things”.[26]

I especially like that last line — which makes the case that, while Feynman had opinions on these matters, he fundamentally believed that in some sense this whole line of inquiry was unscientific, or at least there was not enough data at the time to make a credible scientific argument one way or the other. Ie, “shut up and calculate” was his preferred modus operandi.

That’s a great quote.

And you’re right, I did mean to say “…advances in mathematics have rendered SOME OF those problems soluble by non-iterative means…” As you say, the three-body problem is an obvious outlier.

However, I don’t think we do need ‘an infinite number of logical operations’ to come to many perfect answers to infinite series. We all know that, in certain simple cases, we can determine what the answer will be. For example, 1/2 + 1/4 + 1/8 + 1/16… will come to exactly 1. We can ‘jump’ to the limit without having to even think about the finer and finer terms because we can re-represent that series in a non-infinite manner.

I can think of a couple of other perspectives from which to view infinite series where they don’t involve intractable iteration. The great example of this is the readily computable solutions to many types of differential equations that Laplace transformations provide, which is essentially allows us to take a viewpoint from another dimensionality. Solving electronic circuits using the Argand plane is another example, where classically unsolvable systems turn into simple algebraic problems. And Poincaré maps. And Fourier transforms… All continuous.

This kind of thing crops up regularly in the history of mathematics, and it seems likely (to me, at first cursory glance, at any rate!) that any irreconcilable problems we have now might expect similar perspective shifts from mathematics that will still using continuous models, just of a slightly different nature. A digital model might do more, of course – we’ll have to wait and see.

I expect Alex will provide some examples that I can get my head around which will no doubt clear up my confusion. Already I’m thinking about the digital signal processing mathematics I toyed with in college, whose memory was triggered by this strange duality between discrete and continuous systems, but only where computation is actively involved…

One source of interesting perspective on the validity or lack thereof of continuum based physical theories comes from the work of Gregory Chaitin. His book ‘Meta Math!: The Quest for Omega’ is a nice place to start. The style puts some people off, but I think it’s rather fun, and the content of the book is splendid. In a nutshell, the gist is that we shouldn’t just be doubting continuum based physics, we should be taking a long hard look at continuum mathematics too.

If the universe isn’t continuous, then why do we have continuous mathematics, and the concept of real numbers? Seems like a “real” waste to have such a powerful mathematical concept not actually applying to the “real” world…

Great question!

The answer, I’d propose, is that continuous math isn’t really continuous. It’s a form of symbolic logic for which the symbols correspond to unreachable hypothetical values. Continuous mathematics exists, then, because there are discrete, symbolic patterns that match pretty well onto the behavior of smooth systems, at least so far as we’re able to approximate them.

At some level, this is why smooth math is useful. You can manipulate Pi without ever having to calculate it to infinite precision. It’s very efficient. However, having symmetries in one system that map onto symmetries in another shouldn’t come as a surprise. After all, where would representation theory be without it?

Look really closely at smooth systems, though, and it starts to become clear that approximating is all you can ever do, and that you’re on shaky ground. What you can prove or know only goes so far.

Add to this the fact that in real life, we never encounter a perfect instance of *Pi* any more than we do an instance of *i*, and suddenly the smooth numbers are looking pretty dubious and redundant.